A391423 Positive discriminants of orders of quadratic fields with form class number 1.

5, 8, 13, 17, 20, 29, 37, 41, 52, 53, 61, 68, 73, 89, 97, 101, 109, 113, 116, 125, 137, 149, 157, 164, 173, 181, 193, 197, 212, 233, 241, 244, 269, 277, 281, 292, 293, 313, 317, 337, 349, 353, 356, 373, 388, 389, 397, 409, 421, 433, 436, 449, 452, 457, 461, 500

Offset

1,1

Comments

Not to be confused with A133315, the positive discriminants of orders of quadratic fields with class number 1.

Let h(D) and h+(D) be respectively the class number and the form class number of discriminant D. Since h+(D) = h(D) if the fundamental unit has norm -1 and 2*h(D) otherwise, this sequence consists of D such that h(D) = 1 and that the fundamental unit has norm -1.

Note that the form class group has 2-rank omega(|D|) - t, where omega = A001221, and t = 0 if 32|D, t = 2 if D == 4 (mod 16), t = 1 otherwise (see A391441). As a result, this sequence is exactly the terms in A133315 that are of the form 8, p or 4*p for prime p == 1 (mod 4).

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Prog

(PARI) QFBclassno(D) = qfbclassno(D) * if (D > 0 && quadunitnorm(D) > 0, 2, 1)
isA391423(n) = n%4 <= 1 && !issquare(n) && QFBclassno(n) == 1 \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

Crossrefs

Cf. A306638 (norms of fundamental units of orders of real quadratic fields).

For a list of sequences related to the class numbers of real quadratic fields, see A087048.

Sequence in context: A390233 A097268 A226696 * A285973 A219639 A314426

Adjacent sequences: A391420 A391421 A391422 * A391424 A391425 A391426

Keyword

nonn,changed

Author

Jianing Song, Dec 09 2025

Status

approved